L(s) = 1 | − 1.73i·2-s + (1 − 1.73i)3-s − 0.999·4-s + (1.5 + 0.866i)5-s + (−2.99 − 1.73i)6-s − 1.73i·8-s + (−0.499 − 0.866i)9-s + (1.49 − 2.59i)10-s + (−0.999 + 1.73i)12-s + (2.5 − 2.59i)13-s + (3 − 1.73i)15-s − 5·16-s + 3·17-s + (−1.49 + 0.866i)18-s + (−3 + 1.73i)19-s + (−1.49 − 0.866i)20-s + ⋯ |
L(s) = 1 | − 1.22i·2-s + (0.577 − 0.999i)3-s − 0.499·4-s + (0.670 + 0.387i)5-s + (−1.22 − 0.707i)6-s − 0.612i·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + (−0.288 + 0.499i)12-s + (0.693 − 0.720i)13-s + (0.774 − 0.447i)15-s − 1.25·16-s + 0.727·17-s + (−0.353 + 0.204i)18-s + (−0.688 + 0.397i)19-s + (−0.335 − 0.193i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.548133 - 2.05340i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.548133 - 2.05340i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.5 + 2.59i)T \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 3 | \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.66iT - 37T^{2} \) |
| 41 | \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 - 1.73i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 6.92iT - 59T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 - 1.73i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + (6 + 3.46i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.22591390538241287943977090604, −9.742024094585024207627542155138, −8.418908355405181381093625544004, −7.77106904571123363072722834635, −6.59480973006934592379648538054, −5.89460202440296274178236808090, −4.16025603545610142440732389776, −2.98475326617353434658611669868, −2.20316195685096880467795827332, −1.21302577905034596293648595395,
2.04822601763219732360446813545, 3.64351257075760586848634133217, 4.63166825991413077983803237833, 5.61745489411584586416106827815, 6.37001922387249817557058387479, 7.41412908195781902925890659936, 8.502794122387243407600592933971, 8.987198245281492544597431980022, 9.788635282614945774243050231080, 10.66381790877750573040675626427